Definitions of CDO contract.

e
consider a portfolio of several correlated underlyings (obligors)
.
Each underlying (obligor) pays coupon and may default. A specially created
company (Special Purpose Vehicle) holds such portfolio and issues contracts
called "CDO tranches" that passes both coupons and credit risk on the buyers
of the tranches. We explain below the terms of the contracts.

Let
be default time of underlying
.
Each underlying (=component of the portfolio) is represented by a certain
notional in the portfolio. The fraction of the notional with respect to the
total initial notional is denoted
,
.
We introduce the fractional defaulted
notional
We assume that the recovery at default is the same across the entire
portfolio. We denote such recovery rate by
,
.
Hence, the portfolio loss from the defaults is given
by
Each tranche is parametrized by "attachment" and "detachment" points that
limit the fractional loss of the tranche
as
follows

(Tranche loss)

There are several tranches
,
,
,
,
.
The tranche
is called "equity" tranche, the tranche
is called "senior tranche" and the rest of the tranches are called "mezzanine"
tranches. For purposes of calculation of coupon payments we introduce the
notional of the
tranche:
The senior tranche's notional
is defined by the
relationship

(Senior tranche loss)

The above relationship is necessary to maintain the balance between the coupon
cashflow from the underlying portfolio and the payments to the tranches. For
example, consider the first default in the underlying portfolio. Some notional
defaults and no longer generates coupon. According to the formula
(
Tranche loss
) the equity tranche covers the loss
and the notional of the equity tranche decreases by
.
However, the entire portfolio's notional decreases by
.
Hence, the senior tranche is equipped with the rule that tranche's notional
decreases by
when the equity and mezzanine tranches are taking losses. We confirm that the
formula (
Senior tranche loss
) indeed has
such effect with the following
calculation:
The
represents the situation when the senior tranche is the only remaining tranche
and it covers the entire loss. The
case allows further
transformation
as claimed.

Summary

of CDO contract. Let
and
be the notional ratio and the default time of the obligor
,
.
By definition of
Let
and
be the attachment and detachment points of the tranches. The defaulted
notional is
and the total loss is
where the
is the recovery rate. The loss covered by the tranche
is
and the coupon of the tranche is calculated from the notional of the
tranche
for
and
for the senior tranche
.